Transition in energy spectrum of stably stratified turbulence

Energy spectra for forced stably stratified turbulence are investigated numerically using the Direct Numerical Simulations (DNS) with $1024^3$ grid points. The calculation is done by solving the 3D Navier-Stokes equations under the Boussinesq approximation pseudo-spectrally. Using toroidal-poloidal decomposition (Craya-Herring decomposition), the velocity field is divided into the vortex mode ($\phi_1$) and the wave mode ($\phi_2$). With the initial kinetic energy being zero, the $\phi_1$ spectra as a function of horizontal wave numbers, $k_{\perp}$, first develops a $k_{\perp}^{- 3}$ spectrum for the whole $k_{\perp}$ range, and then $k_{\perp}^{-5/3}$ part appears with rather a sharp transition wave number. Meanwhile the $\phi_2$ spectra shows $k_{\perp}^{-2}$ first, and then $k_{\perp}^{-5/3}$ part appears with the same transition wave number. Spectra for different values of the Brunt-- V\"ais\"al\"a frequency $N^2=1, 10, 50 {\rm and} 100$ are investigated, and we found that the $k_{\perp}^{-3}$ part at the large scale in the $phi_1$ spectra is characterized as 2d turbulence, and that the whole spectrum has the form of $E(k_{\perp})=a\eta_{\perp\phi_1}^{2/3}k_{\perp}^{-3}+C_K \varepsilon_{\perp\phi_1}^{2/3}k_{\perp}^{- 5/3}$ where $\eta_{\perp\phi_1}$ is the horizontal enstrophy dissipation based on the $\phi_1$ energy, and $\varepsilon_{\perp\phi_1}$ is the horizontal $\phi_1$ energy dissipation. Meanwhile we obtain $E(k_{\perp})=b\sqrt{N\varepsilon_{\perp\phi_2}} k_{\perp}^{-2} +C_K\varepsilon_{\perp\phi_2}^{2/3}k_{\perp}^{-5/3}$ for $\phi_2$ where $\varepsilon_{\perp\phi_2}$ is the horizontal $\phi_2$ energy dissipation. For both cases, $C_K\approx 1.2\sim2.0$ is obtained being close to the Kolmogorov constant.